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Glasses, unlike their crystalline counterparts, exhibit low-frequencynonphononic excitations whose frequencies $\omega$ follow a universal$\mathcal{D}\!\left(\omega\right)\!\sim\!\omega^4$ density of states. Theprocess of glass formation generates positional disorder intertwined withmechanical frustration, posing fundamental challenges in understanding theorigins of glassy nonphononic excitations. Here we suggest that \emph{minimalcomplexes} -- mechanically-frustrated and positionally-disordered localstructures -- embody the minimal physical ingredients needed to generateglasslike excitations. We investigate the individual effects of mechanicalfrustration and positional disorder on the vibrational spectrum of isolatedminimal complexes, and demonstrate that ensembles of marginally stable minimalcomplexes yield $\mathcal{D}\!\left(\omega\right)\!\sim\!\omega^4$.Furthermore, glasslike excitation emerge by embedding a single minimal complexwithin a perfect lattice. Consequently, minimal complexes offer a conceptualframework to understand glasslike excitations from first principles, as well asa practical computational method for introducing them into solids.

Internally-stressed and positionally-disordered minimal complexes yield glasslike nonphononic excitations